Computing Matrix, finding kernel and image

by specialnlovin
Tags: bases, image, kernel, linear algebra
 P: 19 Let T: R[x]2$$\rightarrow$$ R[x]3 be defined by T(P(x))=xP(x). Compute the matrix of x with respect to bases {1,x,x2} and {1,x,x2,x3}. Find the kernel and image of T. I know how to do this when given bases without exponents, however I do not know exactly what this is saying and therefore am having a hard time starting it.
 Mentor P: 18,036 Let's compute the first column. You take the first basis element, that is 1. Now you'll have to express T(1)=x in terms of the basis {1,x,x²,x³}. That would be (0,1,0,0). So the first column would consist out of $$\left(\begin{array}{ccc} 0 & ? & ? \\ 1 & ? & ? \\ 0 & ? & ? \\ 0 & ? & ? \end{array} \right)$$ Now for the second and third column, you'll have to express T(x) and T(x²) in terms of the basis {1,x,x²,x³}.
 Mentor P: 18,036 So, if (x,y,z) is in the kernel, then you must have $$\left( \begin{array}{ccc} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right) \left( \begin{array}{c} x\\ y\\ z \end{array} \right) = 0$$ It's not hard to see that this can only be the case iff x=y=z=0